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A NNALES DE L ’I. H. P., SECTION A

R. F. A LVAREZ -E STRADA

The polaron model revisited : rigorous construction of the dressed electron

Annales de l’I. H. P., section A, tome 31, n

o

2 (1979), p. 141-168

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(2)

141

The polaron model revisited:

rigorous construction of the dressed electron

R. F. ALVAREZ-ESTRADA

Departamento de Fisica Teorica, Facultad de Ciencias Fisicas,

Universidad Complutense, Madrid-3.

Vol. XXXI, 2, 1979 Physique theorique.

RESUME. 2014 Le modele du

polar on (c’est-a-dire,

un electron non relativiste

en interaction avec un

champ quantifie

de

phonons optiques)

est etudie

du

point

de vue

mathematique.

Deux methodes sont utilisees pour construire

rigoureusement

1’etat d’un electron habille le

polaron).

La

premiere

est basee sur une combinaison de transformations

d’habillement,

la theorie de

Brillouin-Wigner

et la methode des

approximations

succes-

sives : elle est valable seulement pour des valeurs assez

petites

de la constante

de

couplage.

La deuxieme methode est basee sur :

i)

une version modifiee

et

rigoureuse

de la theorie de

Tamm-Dancoff, specialement

modifiee pour resoudre les

problemes poses

par le modele du

polaron

pour des

grandes energies

des

phonons, ii)

des methodes

iteratives,

de Fredholm et de la theorie de la

dispersion

pour

plusieurs particules,

combinees avec des

theoremes de

point

fixe. II est demontre que cette deuxieme methode est valable pour des valeurs croissantes de la constante de

couplage,

pour les-

quelles

la

premiere

methode n’est

plus valable,

et que, d’autre part, elle fournit des fondaments pour faire des

analyses

non

perturbatives.

ABSTRACT. - The

polaron model,

which describes a non-relativistic quantum

electron, interacting

with a

quantized optical phonon field,

is

studied from a mathematical

standpoint.

Two methods are used in order

to construct

rigorously

the dressed one-electron state

(the polaron).

The

first combines a

dressing transformation,

the

Brillouin-Wigner approach

and the

contraction-mapping principle,

and works

only

for rather small values of the

coupling

constant. The second method is based upon :

i)

a

modified and

rigorous

version of the Tamm-Dancoff

approach, specially adapted

to cope with the

peculiarities

of the

polaron

model at

large phonon

Annales de l’Institut Henri Poincaré-Section A-Vol. XXXI, 0020-2339/1979/141/$ 4,00/

(3)

energies, ii) iterative,

Fredholm and

multiparticle scattering techniques, together withfixed-point

theorems. The second

approach

can be shown

to work for

increasing

values of the

coupling

constant, when the first method breaks

down,

and to

provide

a mathematical basis for non-pertur- bative studies.

1. INTRODUCTION

The

large polaron

model describes a slow conduction electron

interacting

with a

quantized optical phonon

field in an ionic

crystal.

References

[7-9] ] provide comprehensive

accounts of the different

approaches,

results and open

problems

in the

subject.

The

polaron

model is

interesting for,

at

least,

the

following

reasons :

i)

it describes

physical phenomena

in threedimen- sional space,

ii)

it is free of

divergences,

at the level of

perturbation theory,

in the one electron

subspace, iii)

it constitutes an excellent

testing ground

for

estimating

the convergence of

perturbative

methods and

developing non-perturbative techniques.

There is a vast literature about field-theoretic models related more or less to the

polaron

one, from the

standpoint

of

Mathematical

Physics (see [10-?7]).

However, to the author’s

knowledge

and with the

exception

of a rather

short discussion

by

Ginibre in

[2~],

no

rigorous study

of the

polaron

model

seems have been

reported

so far. It turns out that the

polaron

model is

characterized

by

a

phonon

energy and a cut-off or vertex function

v(k) having

a

peculiar dependence

on the threemomentum k

(see

section

2),

which

requires

a

special

treatment.

The purposes of this paper are :

1)

to treat

rigorously

a class of field- theoretic models which includes the standard

large-polaron

one,

and,

very

specially, 2)

to construct

mathematically

the dressed or renormalized one-electron state. The methods to be used and the

corresponding

main

results are the

following.

a)

The

dressing

transformation used

by

Gross

[2~] and

Nelson

[2~] will

be combined with the

Brillouin-Wigner

method

(see,

for

instance, [29 ])

and the

contraction-mapping principle (section

3 and

Appendices A, B).

The main result will be the

rigorous

determination of the dressed electron state via convergent

iteration-perturbation methods,

for rather restricted

values of the

coupling

constant.

b)

A new method will be

presented

in sections

4,

5 and

Appendix C,

which avoids the use of

dressing

transformations. Its

starting point

is a

variant of the Tamm-Dancoff

approach [30-31 ], specially adapted

to cope with the energy and vertex function

characterizing

the models

under consideration

(section 4).

One of the main results is the

rigorous

construction of the dressed electron state for a limited range of values for

Annales de l’Institut Henri Poincaré - Section A

(4)

the

coupling

constant

(subsection

5.

A), through

a new convergent pertur- bation-iteration method which is different from the one

presented

before

in section 3. In the case of the standard

large polaron model,

a numerical

analysis

indicates that convergence now holds for values of the

coupling

constant which may be one to two orders of

magnitude larger

than the upper limit obtained in section 3 for convergence

using dressing

transformations.

Another main result

(subsections

5. Band 5.

C)

is the

rigorous

determination of the dressed electron state for

increasing

values of the

coupling

constant

(actually larger

than the ones

previously

considered in subsection

5.A), by combining perturbation-iteration,

Fredholm and

multi-particle

scat-

tering

methods. This allows for the

possibility

of

exploring,

in a

systematic

and

mathematically

controllable way, the

non-perturbative regime

of

values for the

coupling

constant. A rather

qualitative

numerical

estimate,

which illustrates the last statement, is also

presented.

2. CHARACTERIZATION OF MODELS

We consider a non-relativistic

spinless

quantum electron with bare

mass mo and

position

and threemomentum operators x =

(~), p

=

( p~),

=

1, 2,

3

( [xl, pj = i~~~),

and an indefinite number of

longitudinal optical phonons, regarded

as scalar bosons. Let

I 0 &#x3E;

be the

phonon

vacuum

and

a(k), a + (k)

be the standard destruction and creation operators for

a

phonon

with

(continuously-varying) threemomentum k

and energy

( [a(k), ~(~)] = ~~3~(k - 1~), k

=

!~!).

The bare one-electron state with

threemomentum q

is denoted

by

so that the set of all bare electron-

phonon

states

constitutes a basis for the actual Hilbert space.

By assumption,

the total hamiltonian is :

The total threemomentum operator is

, f

and

v(k)

are,

respectively,

a real

coupling

constant and a

complex

cut-oif function.

They satisfy

the

following assumptions :

1)

Vol. XXXI, n° 2-1979.

(5)

2 lim

(~)

) ~~ ~T7~

either vanishes or is a finite constant,

4)

ex!

--+

v° ,

v0

being

a

non-vanishing complex

constant.

The above

assumptions 1)-4)

define a

slight generalization

of the

physi- cally interesting large polaron model,

characterized

by

&#x3E; 0 and

v ( ) k - ~ k

for any

phonon threemomentum k, -

which

clearly

satisfies

them. We shall

apply

all our latter

developments

to the

large polaron model

with

~’

= 20142014_20142014’ Z7T

2014 ~

, a

being

the usual dimensionless

coupling

mo

constant

[32 ].

03C02

Let be the

subspace

of all

kets 03C8

such that

(Ptot -

= 0 with

.2014

~’‘~o~ in order to avoid unwanted « Cerenkov effects »

[8 ].

-~0 n

The set of all bare states

kn), q03C0 = 03C0 - &#x3E; 03A3kj,

constitutes

a

complete

orthonormal set in

H03C0

with respect to the restricted scalar

product :

In

Eq. (2.4), ~

denotes the usual sum over the n !

permutations

. v( 1 ),...,v(n)

(v(1), ... , v(n)) of (1, ... , n).

Notice that the

ordinary

scalar

product

of the same states in the full

Hilbert space

equals

the restricted one,

given

in

Eq. (2.4),

times a volume

divergent factor, namely, ~~(6).

In the restricted norms of a

state 03C8 belonging

to it and an operator A such that are defined

respectively

as

[(~, ~,)~ ] 1~2

and

11

= least upper bound of

(

varies

through-

out

the

ground

state of H

(the

dressed electron or

polaron)

with

physical

energy E = and small total threemomentum 03C0:

Annales de l’Institut Henri Poincare - Section A

(6)

Both 03C8 +

and E are finite in each order of

perturbation theory,

as

explicit calculations, patterned

after those in

[4] ] [8] ] [33],

show.

Explicitly,

for the

large-polaron

model it has been

conjectured

that

perturbation theory

converges for 0 . 5

(see

D. Pines in

[3 ]).

It is not

straightforward

to

prove the convergence of the whole

perturbation

series

for t/J +, E,

even for

very small a or

f, and,

to author’s

knowledge,

no such

proof

has been

published

so far.

Actually,

the standard arguments

(see,

for

instance, [23]) implying

that

HI

is a small

perturbation

of

Ho

and the usual convergence

proof

for the Neumann

expansion

of

(z - H) -1

1

or t/J +,

for very small

f,

all run into difficulties since

12

= oo.

Thus,

a

rigorous

deter-

mination

of I/J +,

E and estimates of the values of

f

for which convergence holds seem desirable.

3. CONSTRUCTION OF DRESSED ELECTRON USING DRESSING TRANSFORMATION AND BRILLOUIN-WIGNER APPROACH

We

perform

the

dressing

transformation

implemented by

the

unitary

operator exp

T,

where

A

being

a

non-negative

fixed constant and

8(x)

= 1 if x &#x3E;

0, 8(x)

= 0

for x0.

A

lengthy calculation,

similar to those in

[2~-2J], yields :

Notice that - 00 E’ 0. The usefulness of this type of

transformation,

in order to renormalize a field-theoretic model more

singular

than the one

(7)

studied

here,

was established in

[24-25 ].

Its

potential

usefulness for the

polaron

model was

pointed

out

by

Ginibre in

[28 ].

The new hamiltonian H’ can be

given

a mathematical sense,

by extending

to it the

rigorous analysis

of Nelson

[2~] directly. Appendix

A contains

some

rigorous

results which will be useful later in this work.

After the

dressing

transformation, the dressed electron state is

It

belongs

to and fulfills

[H’ - (E - EW+ =

0. We shall

impose

the normalization = 1 and try to construct

c~ +

from

~(~c), by regarding

the latter and all

kl

...

kn)

as

unperturbed

kets and

+ as

perturbation.

One could construct

(z - H,)-1

and

~’+ by using

the

perturbation theory

for

quadratic

forms and the

projection techniques presented

in

[34] ] (see

also

[2~]). However,

it is easier to use the

Brillouin-Wigner approach [29],

which leads

directly

to :

Here, I!

is the unit operator,

Q,,

is the

projector

upon inside

The basic

polycy

will consist in :

i) solving

the linear

Eq. (3 . 8) regarding

E as a parameter,

ii) plugging

the

resulting

solution into

the

right-hand-side

of

Eq. (3 . 9)

and

solving

for E.

We introduce

The series formed

by

all successive iterations of

Eq. (3.8)

can be cast into :

Majorizing Eqs. (3.13)

and

(3.10),

one finds

with x 1

- d 3- k and

0

Ai being

£

given

in

Eq. (A . 3).

Annales de l’Institut Henri Poincare - Section A

(8)

Next,

we consider the

mapping

M : E ~

M(E),

where

M(E)

is

given by

the

right-hand-side

of

Eq. (3.9),

as well as two values

Ei,

= 1, 2, and

their associates

M(EJ

under the

mapping.

Some

majorations yield :

One sees

easily

that for any

complex E, E1, E2

and any real

E, E1, E2

smal-

ler than

due to the

projector ’0 - Moreover, using

results from

Appendix A,

we

give

a bound

for I) gl(E1, E2) 111f

in

Appendix

B

(Eq. (B . 2)),

which shows +

~ for fixed 039B, under the same conditions for E1, E2.

Then,

for

sufficiently

small

f, : a)

there is a domain D in the

complex

-2 -2

E-plane, containing

20142014

and

20142014

+

E’,

which maps into itself under

M,

b)

1 for E inside

D, c) E2)

1 for

E1, E2 belonging

to D.

Then,

the contraction

mapping principle (see,

for

instance, [35 ])

ensures the existence of a

unique

fixed

point

E of

M,

E =

M(E),

which

belongs

to D and can be found

by

successive iterations of

Eq. (3.9).

More-

over, for the fixed

point, ~ +

is

given by

the convergent series

(3.13).

The

analysis

of Nelson

[25 ]

shows that exp

( ± T)

are well defined

unitary

operators.

Then,

the true

polaron

state is

unambiguously given by

Let us summarize some

qualitative

estimates of the convergence condi- tions for the contraction

mapping principle

to

apply,

in the case of the

standard

large polaron

model

(see

section

2)

for 03C0 = 0. One has :

2-1979.

(9)

We have found that the

condition

1 is

roughly

fulfilled

when A = for values of a up to about

0.009,

which is almost

the order of

magnitude

which characterizes semiconductors of type II-VI

[6 ].

The condition 11 1 1 is also

essentially

satisfied under the same conditions.

This upper limit for convergence, a ~

0.009,

is about one to two orders of

magnitude

smaller than the one

previously conjectured (see

D. Pines

in

[3 ]) :

its smallness is due to the

A-dependence

of

~,1,

xl

which,

in turn,

comes from the

dressing

transformation. The new method to be

presented

in the

following

sections will not

rely

on

dressing

transformations and will allow one to

improve

the convergence

conditions,

at least in

principle.

The actual treatment can be

generalized

when an external

homogeneous magnetic

field lz =

(0,0, h), h

&#x3E;

0, along

the

x3-axis

is present. Let

Ah

=

(- ~2,0,0),

e, c be the standard vector

potential (see,

for

instance, [36 ]),

the electron electric

charge

and the

velocity

of

light

in vacuum.

Then,

if

the sustitution

p

~ p - Ah is

done, Eqs. (2.2-3)

remain valid. Notice

c

that the two operators = Pj + =

1,

3 conmute with the actual

H,

but p2 + does not.

Now,

the basic bare states are

(r

=

0, 1, 2, ... )

= - 201420142014

, 3 =

(~1.~3)

and the

H/s being

the standard Hermite .

~ ~ .h "

polynomials.

Let be the

subspace

of all

states 03C8

such that

The set of all bare states

is a

complete

set for with respect to the new restricted scalar

product :

Annoles de l’Institut Henri Poincare - Section A

(10)

Eqs. (3.1-3), (3.5-7)

and the first

Eq. (3.4)

remain valid

provided

that

p ~

p - while the second

equation (3.4)

should be

replaced by

By using

p -~ as well as the restricted norms for vectors and

c

operators and the

quadratic

form

(analogous

to F of

Appendix A)

induced

by (3 . 22)

and the new

H’,

one can show that the

analogue

of

(A. 1) holds,

with the same e1, e2. Let 20142014 in

Eqs. (3. 8-9)

be

replaced respectively 2mo

by

the

ground

state and the energy of the electron in the external

magnetic field, namely, 03C8(03C0;

-

0), -’

20142014 +

(03C03)2 2m0.

After these

substitutions,

the Bril-

2 c

2mo

louin-Wigner equations

remain valid and determine the

ground

state

~+

and the energy E of the dressed electron in the external

magnetic

field.

The

applicability

of the contraction

mapping principle

and the convergence

proofs

can be established as we did before in this

section,

when h = 0.

The

polaron

model in presence of an external

magnetic

field has been studied

previously by

several authors

[~7-~S]:

our brief discussion above tried to

provide

a

rigorous justification

of those works

(at least,

for the

ground state).

4. MODIFIED TAMM-DANCOFF APPROACH AND A USEFUL BOUND

The

polaron

state

~+

can be

expanded

into bare states as :

Thus, J 1l2

is the unnormalized

probability amplitude

for

finding n

Vol. XXXI, 2-1979.

(11)

phonons

in the

polaron,

and is

symmetric

under

interchanges

of

k’s.

The

interest of

having

factored

out -1 ~2

will be

appreciated shortly. Upon replacing

the

expansion (4.1)

into

(H - E)~+ -

0 and

using Eqs. (2.2-4),

one finds the basic recurrence relations

(e1/2n

=

en| 1/2) :

for n =

0, 1, 2, 3,

..., with

b _ 1

= 0. We shall choose the normalization

I 1 l2

= 1 and

introduce,

for later convenience :

Then, Eq. (4.3)

for n = 0 and the set of all

Eqs. (4.3)

for n 1 become

respectively

Here,

W

is the linear operator defined

by

the

right-hand-side

of all

Eqs. (4.3)

.

forn=1,2,3,...

We shall introduce

as well as the

L2-norm

of

bn :

Annales de l’Institut Henri Poincare - Section A

(12)

(no

confusion should arise between the

notations ~bn~2 and,

and the

following

continued fractions :

By recalling Eq. (4 . 7),

the first

Eq. (4. 2)

and

assumptions 1) - 4)

in sec-

tion

2,

one sees

that,

at least for

small and |E|: i)

2n + oo for n

1, ii)

Ln ~ 0 as n oo,

iii)

there exists some R &#x3E; 0 such that

Zr

&#x3E; 0 for any r ~ R.

For n

R,

one proves the

following

bound :

The

inequality (4.9)

can be

proved

in two alternative ways :

1) By integrating

both sides of

Eq. (4. 3) over k1

...

kn

and

using

one derives the three-term recurrence of

inequalities :

The solution of the recurrence of

inequalities (4.12)

is outlined in

Appen-

dix C.

By setting

=

0,

~ Lm

Z’n ~ Zn, II bn ~2

in

(C . 3),

one obtains

(4.9).

2)

Consider the set of all

Eqs. (4. 3)

for n R &#x3E;

0,

cast it into a matrix form similar to

Eq. (4 . 6), containing

in the

inhomogeneous

term

instead of and iterate the

corresponding

matrix system,

thereby generating

an infinite series for R.

By taking

the

L2-norm

Vol. XXXI, 2-1979.

(13)

of the series for

bn

and

majorizing

as in alternative

1),

one finds a

majorizing

series

for ~bn~ 2,

which can be summed into the continued fraction

Zn

in

(4.9),

that

is,

it coincides with the series obtained when

Zn

is

expanded

into power series in

all ~r + 1 ~ 2,

r ~ n. For

brevity,

we omit details.

Notice that to have factored

out |en| -1/2

in the

probability amplitudes

has turned out to be crucial for the

validity

of

(4. 9).

In

fact,

it leads to the

inequalities (4 .10-11 )

with Tn’

which,

in turn,

contain e1 | 1- 1, ( en + 1 1

1

inside the

integrals and, hence,

are finite in

spite

of

assumption 4)

in sec-

tion 2

(and

so on if the derivation of

(4. 9) proceeds through

alternative

2)).

We have been unable to derive a recurrence or a bound similar to

(4.12)

ox

(4 . 9) respectively

for the full

probability amplitudes

due

precisely

to

assumption 4)

and the absence of

factors en 1- 1

inside certain

integrals analogous

to Tm Tn+ 1, which now

diverge.

Remarks. -

i)

A Tamm-Dancoff

approach

to the dressed electron state was also

proposed by

Larsen in an

interesting

paper

[39 ]. However,

he did not factor

out en ~ -1~2

and he did not present any bound or

rigorous study.

ii)

The Tamm-Dancoff

approach

is related to the so-called

N-quantum approximation :

accounts of the latter appear in

[40-41 ].

5. CONSTRUCTION OF DRESSED ELECTRON STATE IN MODIFIED TAMM-DANCOFF APPROACH

We shall construct

rigorously

the

amplitudes

&#x3E;_

1,

and the

polaron

energy E in several cases, for

successively increasing

values

of f.

We assume that

Zr

&#x3E; 0 for any r ~

1,

for

given

small values

of f,

in a certain domain

D 1

for E. The bound

(4 . 9)

for r ~ 2

together

with

II 112

~

(which

can be

proved similarly,

as in alternatives

1 )

or

2)

in section

4)

ensure that the series for all

1,

obtained

by

successive

iterations of the system

(4 . 6)

converges in 1.

Here,

we shall make the

E-dependence

of

bn’s

and T’S

explicit, frequently.

The direct

majoration

of

M I (E) (Eq. (4.5)) gives:

Next,

we shall

study

the

mapping

£

M1 :

E ~

M1(E).

We consider the set of all

Eqs. (4.3)

for n 1

(or

the system

(4.6))

and

Eq. (4.5)

for two

Annales de l’Institut Henri Poincaré - Section A

(14)

values

Ei, E2

such that

Zr(Ei)

&#x3E;

0,r ~ 1,

i =

1, 2,

and subtract them.

By majorizing

as in alternative

1)

of section

4,

one finds:

The solution of the recurrence of

inequalities (5.

A.

3) for bn(E2) [ I 2

Vol. XXXI, 2-1979.

(15)

is also

given

in

Appendix

C.

By combining (5.

A.

2-3), (5.

A.

6-7)

and

(C. 3)

one gets :

with the convention =

1, t 2 . Z2

if t =

1,

2

respectively. Z’n

is

h=2

given by Eq. (C . 2),

with

-+. T~(E2),

-+

T~+ I(E2). By looking

at

the ratio of the

(l

+

l)-th

term over the l-th one in the series on the

right-

hand-side of

Eq. (5.

A

.11)

and

noticing that zn

-+ 0 oo, one shows

that such a series converges.

From

(5. A .1), (4 . 7-8), (5.

A.

8-11), (5.

A.

4)

and

(C . 2)

we see that for

given f

and small

fixed ~ ~ ~,

there is a domain

D2

of values for E such that :

i)

it

contains 03C02/2m0, ii)

all continued fractions

Zn, Zn

are

strictly positive, iii)

it maps into itself under

M 1, iv)

ri2 1 for any

E1, E2 belonging

to it.

Again,

the contraction

mapping principle guarantees

that there exists

a

unique

fixed

point

E =

M1(E) lying

inside

D2,

which can be found

by iterating Eq. (4. 5).

For this fixed

point,

all

amplitudes 1,

are

given by

the convergent series

generated through

the iterations of

(4.6).

We shall consider the standard

large polaron

model

(recall

section

2)

for 7T = 0 and

study numerically

the range of

validity

of the above

rigorous

construction of the dressed electron state, for

increasing

values of the dimensionless

coupling

constant a.

Now,

~n

(n &#x3E; 1)

can be evaluated

explicitly. Then,

in what

follows,

it

will be understood that Ti 1 and

2,

are

replaced respectively by

~1/2

(1 - E)-1~4 . [an/(n -

1 -

(wither

= mo =

1).

We have carried out several types of numerical estimates :

a)

We have studied the

validity

of

Zr

&#x3E; 0 for r ~ 1 and real values of E.

Let

Ei ~0

be some energy

(not necessarily

the

highest one)

such that

Zr(E)

&#x3E; 0 for r ~ 1 and any E ~

Ei.

b)

We have studied the

inequality (5 . A .1)

with

M1(E)

= E and 7r = 0.

For this purpose, we have obtained the solutions

E2

of

which fulfill

E 2 E 1-

Annales de l’Institut Henri Poincaré - Section A

(16)

For a

0.1,

the

rigorous

construction of the dressed electron state outlined in this subsection can be

proved

to converge. We shall not pre- sent detailed numerical results for such a range.

Rather,

we shall

discuss,

in some

detail,

the more

interesting

case a ~ 0.1.

Table I summarizes our numerical results for

E and

We have shown

numerically

that

Zr

&#x3E; 0 for any r ~ 1 ceases to be true for E &#x3E;

E

1 if

oc=0.5,

0 . 6 and 0 . 7.

Moreover,

we have seen that the

inequality (5 . A .1 ),

with

M 1 (E)

= E

and 7T = 0 is satisfied for

E2

E ~

È1.

This means that

E2

can be

regarded

as a lower bound for the exact

polaron

energy. Other

polaron

lower bounds have been

obtained, using

different

techniques,

in

[42 ].

We have studied the

validity

of the

contraction-mapping condition ’12

1

(recall (5.A.11))

for

given a

and

values of E1, E2

in the range

E2 E1, E2 È1.

It turns out that ri2 1 is fulfilled for a ~ 0.3. For a =

0.4,

our estimates indicate that ri2 is close to 1. For a ~ 0.5 we find that ’12 is

appreciably larger

than 1

and,

moreover, that it increases as a does.

Notice that the

polaron self-energy

obtained from standard pertur- bation

theory

up to second order

[8 ], [33 ], EpT,

and

Feynman’s

upper bound for it

[8-9 ], EF, satisfy : i) È1

&#x3E;

EpT

&#x3E;

E2

for o. 1 ~ a ~

0 . 5, ii) EF&#x3E;EpT&#x3E; È1&#x3E; E2

The main conclusion from the above

analysis

is the

following.

The

rigorous

construction of the dressed electron

presented

in this subsection does

certainly

converge for a ~ 0.3.

Quite probably,

it also converges for ri = 0.4. The last statement is motivated

by

our

previous

numerical

findings

and

by

the fact that our

majorations

and the contraction

mapping principle only give

sufficient conditions for convergence. The above conclu- sions

provide

a

partial

answer to the

conjecture by

Pines

[3] commented

in section 2. Thus the present

method,

which avoids the use of

dressing transformations,

allows one to establish the mathematical existence of the

polaron

for values of 03B1 which characterize semiconductors of type 111- V

[6] ]

and which are one and half orders of

magnitude larger

than the upper limit for convergence

0.009)

obtained in section 3. It is uncertain

(and

hard to establish

numerically)

whether the mathematical construc-

tion of this subsection

actually

converges for a &#x3E; 0.4.

Thus,

for a =

0. 5,

the

corresponding

value for

Ei

1 in Table I could allow for convergence

Vol. XXXI, 2-1979.

(17)

to occur :

however,

we remark that

E2

is rather

large

and that 112 is appre-

ciably larger

than 1 in this case. Unless

important

cancellations occur

in the formal solutions obtained

by

successive

iterations,

such convergence

seems more and more doubtful as a increases above

0.4,

since the cor-

responding

values for

È1

and

E2

and 112 become

large.

All this is in agree- ment,

essentially,

with the

conjectures

formulated

by

Pines

[3 ].

5. B. Another

viewpoint

and

study

of the case R = 2.

Throughout

this

subsection,

we shall assume

Zr

&#x3E; 0 for any r ~

2,

for

given f

and a certain domain of values for E.

We cast the set of all

equations (4. 3) for n

2 into a matrix form simi- lar to

(4 . 4), (4 . 6), namely :

Here, B2

is the column vector formed

by all k2),

... ,

bn( k ~

1 ... ...

(that is,

it is obtained

by dropping b 1 ( k ~ )

in

B1),

is a column vector

whose elements vanish

identically

except the first one, which

equals

j

=

1, 2, j ~

i and

W2

is the

corresponding

kernel.

Throughout

this sub-

section,

the bound

(4.9)

remains valid for n

2,

so that the series for each 2 obtained

by iterating Eq. (5. B .1)

converges in

L2-norm.

Upon considering Eq. (4.3)

for n = 1 and

replacing b2 in it by

the

right-

hand-side of

Eq. (4.3)

for n =

2,

one finds the

following

linear

integral equation

for

b 1:

First,

we consider the case " of small

f (so

that

Zi

1 &#x3E; 0 as

well)

and 0 present

a

viewpoint

alternative " and 0

complementary

to that

adopted

0 in subsec-

Annales de l’Institut Henri Poincare - Section A

(18)

tion 5 . A.

Now,

we have a

mapping M 1 (E

-+

M 1 (E)) essentially equi-

valent to the one in subsection 5.

A,

that

is, M 1

is defined

by

the

right-hand-side

of

Eq. (4.5),

and the series formed

by

all iterations of

Eqs. (5. B .1-2).

One has :

but a

slightly

different

for

will be obtained. One finds :

The

inequality (5.

B.

6)

can be

proved by iterating Eq. (5.

B.

2), using

and

summing

the

resulting geometric

series.

Taking L2-norms

in

(5 . B . 3), (5.B.6), using (4 . 9)

for n =

2,

3 and

solving for ~b1 ~2,

one arrives at the

new bound :

Assumptions 1 )-4)

in section 2

imply

the finiteness

of y

1 and One

can obtain another

inequality,

similar to

(5.

A

10),

with a new

positive

constant

r~2,

whose

lengthy expression

will be omitted. As in

previous

cases, the contraction

mapping principle

can be

applied

to the

mapping M

1

when ~ 2 1,

which leads to construct

unique

solutions for E =

M 1 (E)

and all 1.

As f increases,

the conditions

Z

&#x3E; 0 and

1, r~ 2 1

can be

expected

to break down.

Then,

the

inequality (4.9)

would become

meaningless

for n = 1, and the convergence of the series

for b1 obtained by iterating (4 . 6)

Vol. XXXI, 2-1979.

(19)

or

( 5 .

B.

2)

would not be warranted. In order to solve this

problem,

we start

by noticing

that

k 1 )

is a Hilbert-Schmidt kernel

by

virtue of

assumptions 1)-4)

in section 2.

Then,

the modified Fredholm

theory [43] ]

leads to : _

Annales de l’Institut Henri Poincaré - Section A

(20)

According

to Smithies

[43 ],

the

following properties

are valid :

i)

the

series

(5 . B 12), (5 . B 16)

for

ki)

and A

(the

modified first Fredholm minor and Fredholm

determinant, respectively) always

converge,

by

virtue

of the bounds

(5 . B 14-15)

and

(5.B. 18), and ~N~2

+ ~,

ii) by

assu-

ming

4 ~

0,

the

right-hand-side

of

Eq. (5 . B .11) gives

the

unique

solu-

tion of

Eq. (5.B. 2),

which is valid even when the series formed

by

all the

iterations of Eq. (5.

B.

2) diverges.

The bound

(5.

B.

8)

is also

valid, provided

that

Nm(k1, k i )

and

0394m

be

replaced by N( k 1, k i )

and 0394

respectively.

Eqs. (5 . B .1), (5 . B . 3), (5 . B .11)

and the

right-hand-side

of

(4.5)

define

a new

mapping,

also denoted

by M 1 :

E -~

Mi(E).

As in

previous

cases,

the

application

of

fixed-point

theorems to the actual

mapping

allows

to construct solutions for E =

M 1 (E)

and all 1.

We shall

apply

the

developments

of this subsection to the

large polaron

model :

a)

A numerical solution of -

E2

=

zl(E2)’ ~ bl{E2) ~ 112, II ~1(62) 2 being replaced by

the

right-hand-side

of

(5.

B.

8), yields

values for -

E2

which

are

slightly

smaller than those for -

E2 (Table I) : thus,

for a =

0 . 3,

0 . 4 and

0.5,

we get

respectively - E2

=

0.362, 0.551,

0.795. This leads to,

essentially,

the same conclusions as in subsection 5. A : the method based upon

Eq. (4 . 5)

and the series formed

by

the iterations of

Eqs. (5 . B .1-2)

does converge if x ~

0 . 3,

its convergence

being quite plausible

for a =

0 . 4,

and uncertain for a &#x3E; 0.4

(increasingly

doubtful as a

increases).

b)

A numerical

study

for a &#x3E;

0 . 4,

based upon

Eqs. (4 . 5), (5. B .1)

and

Fredholm

theory

is more

complicated.

For this reason, we shall limit ourselves to some

qualitative

estimates. We use

separable approxima-

tions for

kl)

of the type

with n == 1 or 2

(mo

= =

1).

Even if it is hard to estimate the error of

such an

approximation,

we expect the latter to be

moderately adequate

when

ki)

acts upon functions whose

k’1-behavior

ressembles that of the exact

b 1 ( k i ),

in a limited range above 03B1 == 0 . 4.

Here,

one finds

With

these,

we have carried out a numerical

analysis

similar to the one

outlined in

a)

above. Instead of -

E2,

we now find out numerical solu- tions -

63 == 0.515, 0.755,

for II =

0.4, 0.5, respectively (essentially

the

same for both n = 1 and n =

2),

which are

systematically

a bit smaller than -

E2

or -

E2.

This fact could be

perhaps regarded

as a numerical

Vol. XXXI, 2-1979.

(21)

hint of the

reliability

of Fredholm

theory.

We stress that

Zr

&#x3E; 0 for r ~ 2 is

always

fulfilled and believe that

È3

is also a lower bound for the true

polaron

energy.

Here,

the main

qualitative

conclusion is : the method based upon

Eq. (4 . 5),

the series of iterations for

Eq. (5.

B

.1)

and

Eqs. (5.

B

11-18)

can be

expected

to converge for 0.4 ~ a ~ 0 . 5

(while

that of subsection 5. A and the one based upon the iterations

of Eq. (5.

B.

2)

are of uncertain

validity

in such a

range),

at least.

Here,

we assume that

Zr

&#x3E; 0 if r ~

3,

in a certain domain of E and for

given 7T

and somewhat

larger

values

of f .

We allow for

(4 . 9)

to break

down for n = 1. 2 so that we do not expect that either

b 1

or

b2

could be

found

by

successive iterations. The set of all

Eqs. (4.3) for n

3 can be

cast into the matrix system

B 3

== +

B 3 (B3 being

a column vector

obtained from

Eq. (4 . 4), by dropping bi

1 and

b2, etc.) : by iterating

the

latter,

one obtains a series for

3,

which converges in

L2-norm,

as the bound

(4.9)

is

meaningful

for n 3.

We shall concentrate in

displaying

and

solving

a new

difficulty regard- ing b2,

which did not appear when R = 2,

omitting

unnecessary details.

By considering Eq. (4.3)

for n = 2 and

replacing

in it

b3 by

the

right-

hand-side of

Eq. (4. 3)

for n =

3,

one finds the linear

integral equation :

where j

=

1, 2, j

~ i in

Eqs. (5 .

C .

2-3)

and 62 is

given

in

Eq. (5 .

B .

5). Sym-

2

bolically,

2 we shall rewrite

Eq. (5 . C .1)

as

b2

= +

Ii

i= 1

b2.

The

kernel

03A3li

is not Hilbert-Schmidt in all threemomenta

kl, k2, ki, k2,

i= 1

due to the structure of the

right-hand-side ofEq. (5. C.1),

so that the modified

Annales de l’Institut Henri Poincare - Section A

(22)

Fredholm

theory

cannot be used to solve

(5.

as it stands. To solve this

difficulty,

we shall

apply

rearrangement

techniques typical

of multi-

particle scattering theory [44 ]. Using symbolic

notation

partially,

the basic

new

equations

read :

Notice that either

by iterating

eqs.

(5 .

C .

5)

and the second

Eq. (5 .

C .

4)

and

inserting

the

resulting

series into the first

Eq. (5 .

C .

4),

or

by iterating directly Eq. (5.

C

.1),

one arrives at the same formal series. The main pro-

perties

of

Eqs. (5 .

C .

4-5)

are:

a)

The kernel

li

is Hilbert-Schmidt in

kz, k’i,

for fixed i.

Then,

the solution of

Eq. (5 .

C .

5)

is

given by

the modified Fredholm

theory,

that

is, by performing

suitable

replacements

in

Eqs. (5 .

B

.11-17). Sym- bolically,

we writte:

03B6i

-

t. Z + 1 0394, N.l. L i7

i =

1

, 2. Notice that

0.

I

depends

1B.

l

only

on i .

b)

We notice that the kernel i #

j,

is defined for any cp =

k2)

as follows:

and so on for

ç 2. ç 1.

In what

follows,

we shall assume that

and so on

for ~~ 12~k1~’~ !!2. 12,1.

Notice

that 1~,2

+

00, [2,1

i + oo. We

can prove that

- l.

+

is

a bounded operator.

Vol. XXXI, n° 2-1979.

(23)

In

fact, by using

the

analogues

of

(5 . B .12)

and

(5 . B .14),

some direct

majorations give :

The series in

(5.

C.

6)

converge, and the

proof

for the

remaining

contri-

butions to is similar.

Moreover,

since

one finds

easily

a finite bound for

II ~

+

112

in terms of

II bl 112 and ~b4~2.

c)

The kernel is Hilbert-Schmidt :

and

similarly

for

Ç2Çl.

Let us sketch a direct

proof. By using

the

analogues

+00

of

(5.

B

.12), namely, N, = ~ +

i == 1,

2,

one finds:

Annales de l’Institut Henri Poincare - Section A

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